Calculating the energy and wavelength of electron transitions in a one–electron (bohr) system
What is the energy (in joules) and the wavelength (in meters) of the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit with
n = 4 to the orbit with
n = 6? In what part of the electromagnetic spectrum do we find this radiation?
Solution
In this case, the electron starts out with
n = 4, so
n
_{1} = 4. It comes to rest in the
n = 6 orbit, so
n
_{2} = 6. The difference in energy between the two states is given by this expression:
$\text{\Delta}E={E}_{1}-{E}_{2}=2.179\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-18}}\left(\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{1}^{2}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{{n}_{2}^{2}}\phantom{\rule{0.2em}{0ex}}\right)$
$\text{\Delta}E=2.179\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-18}}\left(\phantom{\rule{0.2em}{0ex}}\frac{1}{{4}^{2}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{{6}^{2}}\phantom{\rule{0.2em}{0ex}}\right)\phantom{\rule{0.2em}{0ex}}\text{J}$
$\text{\Delta}E=2.179\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-18}}\left(\phantom{\rule{0.2em}{0ex}}\frac{1}{16}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{36}\phantom{\rule{0.2em}{0ex}}\right)\phantom{\rule{0.2em}{0ex}}\text{J}$
$\text{\Delta}E=7.566\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-20}}\phantom{\rule{0.2em}{0ex}}\text{J}$
This energy difference is positive, indicating a photon enters the system (is absorbed) to excite the electron from the
n = 4 orbit up to the
n = 6 orbit. The wavelength of a photon with this energy is found by the expression
$E\text{=}\phantom{\rule{0.2em}{0ex}}\frac{hc}{\lambda}.$ Rearrangement gives:
$\lambda =\phantom{\rule{0.2em}{0ex}}\frac{hc}{E}$
$\begin{array}{}\\ \\ =\left(6.626\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-34}}\phantom{\rule{0.2em}{0ex}}\overline{)\text{J}}\phantom{\rule{0.2em}{0ex}}\overline{)\text{s}}\right)\phantom{\rule{0.4em}{0ex}}\times \phantom{\rule{0.3em}{0ex}}\frac{2.998\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}{\overline{)\text{s}}}^{\mathrm{-1}}}{7.566\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-20}}\phantom{\rule{0.2em}{0ex}}\overline{)\text{J}}}\phantom{\rule{0.2em}{0ex}}\\ =2.626\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}\phantom{\rule{0.2em}{0ex}}\text{m}\end{array}$
From
[link] , we can see that this wavelength is found in the infrared portion of the electromagnetic spectrum.
Check your learning
What is the energy in joules and the wavelength in meters of the photon produced when an electron falls from the
n = 5 to the
n = 3 level in a He
^{+} ion (
Z = 2 for He
^{+} )?
Answer:
6.198
$\times $ 10
^{–19} J; 3.205
$\times $ 10
^{−7} m
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Bohr’s model of the hydrogen atom provides insight into the behavior of matter at the microscopic level, but it is does not account for electron–electron interactions in atoms with more than one electron. It does introduce several important features of all models used to describe the distribution of electrons in an atom. These features include the following:
Of these features, the most important is the postulate of quantized energy levels for an electron in an atom. As a consequence, the model laid the foundation for the quantum mechanical model of the atom. Bohr won a Nobel Prize in Physics for his contributions to our understanding of the structure of atoms and how that is related to line spectra emissions.
Bohr incorporated Planck’s and Einstein’s quantization ideas into a model of the hydrogen atom that resolved the paradox of atom stability and discrete spectra. The Bohr model of the hydrogen atom explains the connection between the quantization of photons and the quantized emission from atoms. Bohr described the hydrogen atom in terms of an electron moving in a circular orbit about a nucleus. He postulated that the electron was restricted to certain orbits characterized by discrete energies. Transitions between these allowed orbits result in the absorption or emission of photons. When an electron moves from a higher-energy orbit to a more stable one, energy is emitted in the form of a photon. To move an electron from a stable orbit to a more excited one, a photon of energy must be absorbed. Using the Bohr model, we can calculate the energy of an electron and the radius of its orbit in any one-electron system.